EE113: DIGITAL SIGNAL PROCESSING Midterm 1 Practice Problems
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EE113: DIGITAL SIGNAL PROCESSING
Midterm 1 Practice Problems
Problem 1
Compute the convolution of x[n] and h[n], y[n] = x[n] * h[n].
a) x[n] = { -3 , 7, 4} and h[n] = { 4 , 3, 2, 1} n0 n0
b) x[n] = u[n] - u[n - 7] and h[n] = (0.9)nu[n]
Problem 2
a) Consider a discrete-time complex signal x[n] = A[n]ejφ[n], where A[n] = lx[n]l and φ[n] is the phase of the signal x[n].
Derive the relationship between A[n] and A[-n], and φ[n] and φ[-n] when the signal is
i) conjugate symmetric
ii) conjugate antisymmetric
b) Now consider a discrete-time complex signal x[n] = a[n] + jb[n], where a[n] is the real part and b[n] is the imaginary part of x[n]. Is a[n] and b[n] odd or even when the signal x[n] is:
i) conjugate symmetric
ii) conjugate antisymmetric
Problem 3
Assume x[n] has nonzero samples only in the interval -N1 5 n 5 N2 . Generally, over what interval of time will the following sequence have non-zero samples:
y[n] = x[n] * x[n]
Problem 4
Prove the distributive property of the periodic convolution:
[n] 8 (y˜[n] + [n]) = [n] 8 y˜[n] + [n] 8 [n]
Problem 5
Consider the following system: y[n] = k(n)=0 x[k]
(a) Is the system linear? Prove your answer.
(b) Is the system time-invariant? Prove your answer.
(c) Is the system causal? Prove your answer.
(d) Is the system BIBO stable? Prove your answer. (_乞脯t_ :Оu 注Hy 脯eed tО use tr乞H脯夕≥e 乞脯eζuH≥乞ty_ lx + yl < lxl + lyl)
Problem 6
Consider a signal x[n] that has a DTFT depicted in the figure below in the range [-π, π].
Find the expression for the DTFT of the signals below:
(a) x1 [n] = nx[n - 1]
(b) x2 [n] = ej (x[n] * x[n])
(b) xo [n], the odd part of x[n]
Problem 7
Consider the system composed of parallel connection of two LTI systems.
(a) If unit-step response of the equivalent system (the response when the input is a unit-
step function) is y[n] = r[n + 1] - r[n - 1] and h1 [n] = u[n] - 2u[n - 1] + u[n - 2], find and sketch h2 [n].
(b) Find the equivalent impulse response of the system heq [n]. The equivalent response is defined by the following relation: y[n] = x[n] * heq [n].
(c) Find the response of the system y[n] for x[n] shown in the figure below.
Problem 8
Let [n] be a periodic signal with period N . Its DTFS representation is given by
N-1
[n] = k ej kn ,
k=0
where k are the DTFS coefficients.
Show that if [n] is a complex signal and conjugate symmetric (* [n] = [-n]), then Im{k } = 0.
Problem 9
Consider a periodic signal [n] signal with one if its periods shown in the figure below. Cal- culate its DTFS coefficients k .
2022-11-26